Elliptic solutions of Boussinesq type lattice equations and the elliptic Nth root of unity

TL;DR

This paper develops elliptic function solutions for lattice Boussinesq systems using a direct linearisation approach, introducing the novel concept of elliptic Nth roots of unity to facilitate reductions from elliptic KP equations.

Contribution

It introduces the elliptic Nth root of unity concept and constructs elliptic solutions for lattice Boussinesq systems via a new linearisation scheme from elliptic KP equations.

Findings

Established elliptic solutions for lattice Boussinesq systems.

Introduced the elliptic Nth root of unity concept.

Derived elliptic N-soliton solutions.

Abstract

We establish an infinite family of solutions in terms of elliptic functions of the lattice Boussinesq systems by setting up a direct linearisation scheme, which provides the solution structure for those equations in the elliptic case. The latter, which contains as main structural element a Cauchy kernel on the torus, is obtained from a dimensional reduction of the elliptic direct linearisation scheme of the lattice Kadomtsev-Petviashvili equation, which requires the introduction of a novel technical concept, namely the "elliptic cube root of unity". Thus, in order to implement the reduction we define, more generally, the notion of {\em elliptic $N^{\rm th}$ root of unity}, and discuss some of its properties in connection with a special class of elliptic addition formulae. As a particular concrete application we present the class of elliptic $N$-soliton solutions of the lattice Boussinesq systems.